I am using a two-column design and would like the last page (when there is not enough material to fill the whole page) to be laid out such that the material flows into just the first column until it is completely filled and then into the second column only if the first column is full:

xxx xxx
xxx xxx
xxx
xxx

rather than

xxx xxx
xxx xxx
xxx xxx

and

xxx
xxx
xxx
xxx

rather than

xxx xxx
xxx xxx

What should I do to prevent the automatic balancing of column depth on the final page?

You can use the column-fill property with "auto" to get unbalanced columns.